Abstract

Let $A=E_{\mathfrak{1}}\times E_{\mathfrak{2}}$ be the product of two elliptic curves over $\mathbb{Q}$, each having a rational $\mathfrak{5}$-torsion point $P_i$. Set $B:=A∕\langle \left(P_{\mathfrak{1}},P_{\mathfrak{2}}\right)\rangle$. In this paper we give an algorithm to decide whether the order of the Tate-Shafarevich group of the abelian surface $B$ is square or five times a square, under the assumptions that we can find a basis for the Mordell-Weil groups of $E_{\mathfrak{1}}$ and $E_{\mathfrak{2}}$ and that the Tate-Shafarevich groups of $E_{\mathfrak{1}}$ and $E_{\mathfrak{2}}$ are finite.

We considered all pairs $\left(E_{\mathfrak{1}},E_{\mathfrak{2}}\right)$ with prescribed bounds on the conductor and the coefficients in a minimal Weierstrass equation. In total we considered around $\mathfrak{2}\mathfrak{0}.\mathfrak{0}$ million abelian surfaces, of which $\mathfrak{4}\mathfrak{9}.\mathfrak{1}\mathfrak{6}\%$ have Tate-Shafarevich groups of nonsquare order.

Keywords

Mathematical Subject Classification 2010

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